Question

把数字 $1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$ , $9$ 填入一个 $3\times3$ 方格阵列，每格一个数，使得任意两个连续数字所在方格都共边。四个角上的数字之和为 $18$ 。中心格中的数字是多少？

All the numbers $1,$ $2,$ $3,$ $4,$ $5,$ $6,$ $7,$ $8,$ $9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18.$ What is the number in the center?

$\ 5$

$\ 6$

$\ 7$

$\ 8$

$\ 9$

答案：C

知识点：奇偶性逻辑推理

难度评级：1420

解答：

把方阵像棋盘一样染色，使中心和四个角是一种颜色，四个边格是另一种颜色。相邻整数占据相邻格，所以路径 $1,2,\ldots,9$ 的颜色交替。因此五个奇数全在一个颜色类中，四个偶数全在另一个颜色类中。

由中心和四个角组成的颜色类有五个格子，所以其中放的是五个奇数。因此四角与中心之和为 $1+3+5+7+9=25.$

由于四个角的和为 $18,$ 中心的数为 $25-18=7.$

所以正确答案是 C。

Color the array like a checkerboard, with the center and four corners one color and the four edge squares the other color. Consecutive numbers occupy adjacent squares, so the path $1,2,\ldots,9$ alternates colors. Therefore all five odd numbers occupy one color class and all four even numbers occupy the other.

The color class consisting of the center and four corners has five squares, so it contains the five odd numbers. Hence the sum of the corners and center is $1+3+5+7+9=25.$

Since the corners have a sum of $18,$ the center has a value of $25-18=7.$

Thus, the correct answer is C .

2016 AMC 10B 第 15 题

答案：C

难度评级：1420

解答：

其他年份的第 15 题